A common failure in the provision of connectionless service in a network of communications nodes arranged in an arbitrary topology is the inability to route, in an efficient manner, a broadcast message from a source node to each of the other nodes. For example, in one prior arrangement, broadcasting is done by arranging the source node so that it sends a copy of the broadcast message to each of the other nodes in the network. Although this arrangement achieves the desirable goal of sending only a single copy of a broadcast message to each node, it, nevertheless, does not make efficient use of the network bandwidth, since multiple copies of the broadcast message may travel over the same route(s) to reach intended destinations.